「霞む花一枚」 is a very rhetorical expression which is unlikely to be seen except in lyrics.
Grammatically, 「霞む花一枚」 is not strongly linked to something in the previous line. It's a noun phrase, forming one sentence by its own. "A petal from a misty flower."
I can't logically explain what "霞む花" (literally, hazy or misty flower) means here. There is a plant called カスミソウ but it probably is irrelevant. And there is also a word 花がすみ, which means full-bloomed cherry blossoms looking like mist, but full bloom is definitely not what she wants to say in this nostalgic situation.
It seems to me that the person here is reminded of old memories by seeing the flower. So it may be a kind of metaphor for the "vagueness" of the memory.
"枚" is not used to count a flower. "1枚の花" is incorrect in plain Japanese. Instead, "枚" is used to count petals (1枚の花びら). So the best way to understand "霞む花 一枚" is to assume an imaginary petal: "霞む花（の花びらが）一枚". That's what the translator did.